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On fractional integration formulae for Aleph functions. (English) Zbl 1242.33021
Summary: This paper is devoted to the study of a new special function, which is called Aleph function, according to the symbol used to represent this function. This function is an extension of the \(I\)-function, which itself is a generalization of the well-known and familiar \(G\)- and \(H\)-functions in one variable. In this paper, a notation and a complete definition of the Aleph function will be presented. Fractional integration of Aleph functions, in which the argument of the Aleph function contains a factor \(t^{\lambda }(1 - t)^{\mu }, \lambda , \mu > 0\), will be investigated. The results derived are of most general character and include many results given earlier by various authors including A. A. Kilbas [Fract. Calc. Appl. Anal. 8, No. 2, 113–126 (2005; Zbl 1144.26008)], A. A. Kilbas and M. Saigo [Fukuoka Univ. Sci. Rep. 28, No. 2, 41–51 (1998; Zbl 0923.33006)] and L. Galué [Int. J. Appl. Math. 10, No. 3, 255–267 (2002; Zbl 1036.33002)] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature.

33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
26A33 Fractional derivatives and integrals
Full Text: DOI
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